Octagar family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
Octagar temperament is the 7-limit [[Planar_Temperament|planar temperament]] tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-22 13:18:34 UTC</tt>.<br>
: The original revision id was <tt>247708793</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Octagar temperament is the 7-limit [[planar temperament]] tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.


===Vital statistics===
===Vital statistics===
[[Comma]] c = 4000/3969
[[Comma|Comma]] c = 4000/3969


7-limit minimax: 3 and 7 1/6c sharp, 5 just
7-limit minimax: 3 and 7 1/6c sharp, 5 just
[|1 0 0 0&gt;, |5/6 1/3 1/2 -1/3&gt;,  
[|1 0 0 0&gt;, |5/6 1/3 1/2 -1/3&gt;,  
|0 0 1 0&gt;, |5/6 -2/3 1/2 2/3&gt;]
|0 0 1 0&gt;, |5/6 -2/3 1/2 2/3&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 5/4
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 5/4


9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
[|1 0 0 0&gt;, |5/8 1/2 3/8 -1/4&gt;,  
[|1 0 0 0&gt;, |5/8 1/2 3/8 -1/4&gt;,  
|0 0 1 0&gt;, |5/4 -1 3/4 1/2&gt;]
|0 0 1 0&gt;, |5/4 -1 3/4 1/2&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7


Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
Angle(63/50, 21/20) = 97.743 cents
Angle(63/50, 21/20) = 97.743 cents
Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
[[edo|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
 
[[EDO|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
 
Badness: 0.000216
Badness: 0.000216


[[Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5
[[Projection_pair|Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5


===[[Hobbits|Hobbit bases]]===
===[[Hobbits|Hobbit bases]]===
Line 33: Line 34:


12: 50/49, 256000/250047
12: 50/49, 256000/250047
15: 256000/250047, 1029/1000
15: 256000/250047, 1029/1000
23: 12800000/12252303, 107163/102400
23: 12800000/12252303, 107163/102400


==Nakika==
==Nakika==
Commas: 100/99, 245/242
Commas: 100/99, 245/242
Associated linear temperament: [[Tetracot family|octacot]]
 
Associated linear temperament: [[Tetracot_family|octacot]]
 
Lattice basis: 11/7 0.798 22/21 0.906
Lattice basis: 11/7 0.798 22/21 0.906
Angle(11/7, 22/21) = 97.747 degrees
Angle(11/7, 22/21) = 97.747 degrees
Map to lattice: [&lt;0 1 2 1 2|, &lt;0 -1 0 2 2|]
Map to lattice: [&lt;0 1 2 1 2|, &lt;0 -1 0 2 2|]


Map: [&lt;1 0 1 4 4|, &lt;0 1 0 -2 -2|, &lt;0 0 2 3 4|]
Map: [&lt;1 0 1 4 4|, &lt;0 1 0 -2 -2|, &lt;0 0 2 3 4|]
EDOs: 12, 15, 26, 29, 41
EDOs: 12, 15, 26, 29, 41
Badness: 0.000539
Badness: 0.000539


Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7
Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7


Scales: [[nakika12]]</pre></div>
Scales: [[nakika12|nakika12]]     [[Category:family]]
<h4>Original HTML content:</h4>
[[Category:octagar]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Octagar family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Octagar temperament is the 7-limit &lt;a class="wiki_link" href="/planar%20temperament"&gt;planar temperament&lt;/a&gt; tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.&lt;br /&gt;
[[Category:planar]]
&lt;br /&gt;
[[Category:theory]]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 4000/3969&lt;br /&gt;
&lt;br /&gt;
7-limit minimax: 3 and 7 1/6c sharp, 5 just&lt;br /&gt;
[|1 0 0 0&amp;gt;, |5/6 1/3 1/2 -1/3&amp;gt;, &lt;br /&gt;
|0 0 1 0&amp;gt;, |5/6 -2/3 1/2 2/3&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 5/4&lt;br /&gt;
&lt;br /&gt;
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
[|1 0 0 0&amp;gt;, |5/8 1/2 3/8 -1/4&amp;gt;, &lt;br /&gt;
|0 0 1 0&amp;gt;, |5/4 -1 3/4 1/2&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 9/7&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605&lt;br /&gt;
Angle(63/50, 21/20) = 97.743 cents&lt;br /&gt;
Map to lattice: [&amp;lt;0 -1 -2 -1|, &amp;lt;0 -1 0 2|]&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo"&gt;121&lt;/a&gt;, &lt;a class="wiki_link" href="/162edo"&gt;162&lt;/a&gt;, &lt;a class="wiki_link" href="/189edo"&gt;189&lt;/a&gt;, &lt;a class="wiki_link" href="/215edo"&gt;215&lt;/a&gt;, &lt;a class="wiki_link" href="/230edo"&gt;230&lt;/a&gt;&lt;br /&gt;
Badness: 0.000216&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;s: 5 3969/800 7 27783/4000 to 2.3.7/5&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Hobbit bases"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/Hobbits"&gt;Hobbit bases&lt;/a&gt;&lt;/h3&gt;
{2, 3, 7/5} subgroup&lt;br /&gt;
&lt;br /&gt;
12: 50/49, 256000/250047&lt;br /&gt;
15: 256000/250047, 1029/1000&lt;br /&gt;
23: 12800000/12252303, 107163/102400&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Nakika"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Nakika&lt;/h2&gt;
Commas: 100/99, 245/242&lt;br /&gt;
Associated linear temperament: &lt;a class="wiki_link" href="/Tetracot%20family"&gt;octacot&lt;/a&gt;&lt;br /&gt;
Lattice basis: 11/7 0.798 22/21 0.906&lt;br /&gt;
Angle(11/7, 22/21) = 97.747 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 2 1 2|, &amp;lt;0 -1 0 2 2|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 4 4|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 2 3 4|]&lt;br /&gt;
EDOs: 12, 15, 26, 29, 41&lt;br /&gt;
Badness: 0.000539&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/nakika12"&gt;nakika12&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

Octagar temperament is the 7-limit planar temperament tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.

Vital statistics

Comma c = 4000/3969

7-limit minimax: 3 and 7 1/6c sharp, 5 just

[|1 0 0 0>, |5/6 1/3 1/2 -1/3>, |0 0 1 0>, |5/6 -2/3 1/2 2/3>]

Eigenmonzos: 2, 7/6, 5/4

9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp

[|1 0 0 0>, |5/8 1/2 3/8 -1/4>, |0 0 1 0>, |5/4 -1 3/4 1/2>]

Eigenmonzos: 2, 5/4, 9/7

Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605

Angle(63/50, 21/20) = 97.743 cents

Map to lattice: [<0 -1 -2 -1|, <0 -1 0 2|]

EDOs: 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230

Badness: 0.000216

Projection pairs: 5 3969/800 7 27783/4000 to 2.3.7/5

Hobbit bases

{2, 3, 7/5} subgroup

12: 50/49, 256000/250047

15: 256000/250047, 1029/1000

23: 12800000/12252303, 107163/102400

Nakika

Commas: 100/99, 245/242

Associated linear temperament: octacot

Lattice basis: 11/7 0.798 22/21 0.906

Angle(11/7, 22/21) = 97.747 degrees

Map to lattice: [<0 1 2 1 2|, <0 -1 0 2 2|]

Map: [<1 0 1 4 4|, <0 1 0 -2 -2|, <0 0 2 3 4|]

EDOs: 12, 15, 26, 29, 41

Badness: 0.000539

Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7

Scales: nakika12

Retrieved from "https://en.xen.wiki/index.php?title=Octagar_family&oldid=2859"